Signature or simplex

Let us now consider operator $\hat{\cal{O}}$ which is even or odd with respect to one of the six linear D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators, signatures $\hat{\cal{R}}_{k}$ or simplexes $\hat{\cal{S}}_{k}$, k=x,y,z, which have squares equal to minus unity, ( $\hat{\cal{X}}^2$= $\bar{\cal{E}}$= $-\hat{\cal{E}}$, where $\hat{\cal{X}}$ denotes one of them), i.e.,

$$\hat{\cal{X}}^\dagger\hat{\cal{O}}\hat{\cal{X}}= \epsilon_X\hat{\cal{O}}, \qquad \epsilon_X=\pm 1.$$ (18)

Since every operator $\hat{\cal{X}}$commutes with the time-reversal $\hat{\cal{T}}$, one can always choose a basis in which Eq. (9) holds, and

$$\hat{\cal{X}}\vert\mbox{{\boldmath {$n$ }}}\,\zeta\rangle = i\zeta\vert\mbox{{\boldmath {$n$ }}}\,\zeta\rangle,$$ (19)

where again $\zeta$=$\pm$1, so r=$\zeta{i}$=$\pm{i}$ is the signature (for $\hat{\cal{X}}$= $\hat{\cal{R}}_{k}$) or simplex s=$\zeta{i}$=$\pm{i}$ (for $\hat{\cal{X}}$= $\hat{\cal{S}}_{k}$) quantum number. Table 4 lists such bases constructed for the HO states |n_xn_yn_z,s_z= $\pm\frac{1}{2}\rangle$. A similar construction is possible for any other single-particle basis. Note that one can arbitrarily change the phases of states $\vert\mbox{{\boldmath {$n$ }}}\,\zeta$=$+1\rangle$, and still fulfill Eqs. (9) and (19); we shall use this freedom in Secs. 3.1.8 and 3.2.

From Eqs. (18) and (19) one gets:

$$\langle \mbox{{\boldmath {$n$ }}}\,\zeta\vert\hat{\cal{O}}\... ...\hat{\cal{O}}\vert \mbox{{\boldmath {$n$ }}}'\,\zeta'\rangle,$$ (20)

so the matrix ${\cal{O}}$ has the form

$$\left(\begin{array}{cc} A & 0 \\ 0 & B \end{array} \rig... ...egin{array}{cc} 0 & Y \\ Y^\dagger & 0 \end{array} \right)$$ (21)

for $\epsilon_X$=+1 and $\epsilon_X$=-1, respectively, with A and B hermitian, and Y arbitrary complex matrix. Note that in order to diagonalize ${\cal{O}}$ one only needs to diagonalize: (i) for $\epsilon_X$=+1, two hermitian matrices with complex, in general, submatrices in twice smaller dimension, which gives real eigenvalues with no additional restrictions, or (ii) for $\epsilon_X$=-1, one hermitian matrix $Y^\dagger Y$, again with complex submatrices in twice smaller dimension, which gives pairs of real non-zero eigenvalues with opposite signs.

 

Table 4: Eigenstates $\vert n_xn_yn_z\,\zeta\rangle_k$ of the signature or simplex operators, $\hat{\cal{R}}_{k}$ or $\hat{\cal{S}}_{k}$ for k=x, y, or z, [cf. Eqs. (9) and (19)], determined for the harmonic oscillator states |n_xn_yn_z,s_z= $\pm\frac{1}{2}\rangle$. Symbols (N_x,N_y,N_z) refer to (n_x,n_y,n_z) for $\hat{\cal{S}}_{k}$operators, and to (n_y+n_z,n_x+n_z,n_x+n_y) for $\hat{\cal{R}}_{k}$ operators. Phases of eigenstates are fixed so as to fulfill condition (41).

k	$\zeta$	$\vert n_xn_yn_z\,\zeta\rangle_k$
x	+1		$\frac{1}{\sqrt{2}}$	(|nxnynz,sz= $\frac{1}{2}\rangle$	-	(-1)Nx	|nxnynz,sz= $-\frac{1}{2}\rangle\,)$
x	-1		$\frac{(-1)^{N_x}}{\sqrt{2}}$	(|nxnynz,sz= $\frac{1}{2}\rangle$	+	(-1)Nx	|nxnynz,sz= $-\frac{1}{2}\rangle\,)$
y	+1		$\frac{i^{N_y}}{\sqrt{2}}$	(|nxnynz,sz= $\frac{1}{2}\rangle$	-	i(-1)Ny	|nxnynz,sz= $-\frac{1}{2}\rangle\,)$
y	-1		$\frac{i^{N_y-1}}{\sqrt{2}}$	(|nxnynz,sz= $\frac{1}{2}\rangle$	+	i(-1)Ny	|nxnynz,sz= $-\frac{1}{2}\rangle\,)$
z	+1	$+i^{N_z}\exp(-i\frac{\pi}{4})\, \vert n_xn_yn_z,s_z$= $+\frac{1}{2}(-1)^{N_z+1}\rangle$
z	-1	$-i^{N_z}\exp(+i\frac{\pi}{4})\, \vert n_xn_yn_z,s_z$= $-\frac{1}{2}(-1)^{N_z+1}\rangle$

Comparing result (21) with that obtained in Sec. 3.1.2, one sees that the antilinear symmetries allow for using real matrices, while linear symmetries give special block-diagonal forms for complex matrices.